Reversing coffee-ring effect by laser-induced differential evaporation

ABSTRACT

Disclosed are methods, devices and systems to cause differential evaporation of micro-droplets in a scalable fashion. In some embodiments, an attenuated laser is used to focus an attenuated laser power to the center of an aqueous solution droplet, producing a differential evaporative flux profile that is peaked at the droplet apex. The laser-induced differential evaporation described herein is a breakthrough in the enrichment and focused deposition of water-soluble molecules such as nucleic acids, proteins, inks, and other small molecules. Disclosed is a general solution to remove the “coffee-ring effect”, ubiquitous in the drying process of aqueous droplets that causes many adverse outcomes. The disclosed techniques enable new paradigms in liquid biopsy combinational analysis, microarray fabrication, and ink-jet printing.

CROSS-REFERENCE TO RELATED APPLICATION

This patent document claims priority to and benefits of U.S. ProvisionalPatent Application No. 62/656,922, entitled “REVERSING COFFEE-RINGEFFECT BY LASER-INDUCED DIFFERENTIAL EVAPORATION,” filed on Apr. 12,2018. The entire content of the above patent application is incorporatedby reference as part of the disclosure of this patent document.

TECHNICAL FIELD

The present document is related to inducing evaporation of droplets.

BACKGROUND

The drying of a droplet of water carrying colloidal particles naturallygives rise to non-homogenous deposition pattern, with most of particlesmigrating to the edge of the droplet, forming the well-known“coffee-ring” deposition pattern. This non-uniform deposition has posedtechnical challenges in ink-jet printing, DNA/RNA and protein microarraymanufacturing, and in combinational liquid biopsy analysis methods suchas fluorescent microarray, infrared spectroscopy, and Ramanspectroscopy, as well as other applications. New techniques are neededto eliminate the “coffee-ring effect” to increase the quality inprinting, microarray fabrication and analysis, and in advancing liquidbiopsy analysis sensitivity and accuracy.

SUMMARY

In one example aspect, an apparatus is disclosed. The apparatus issuitable for causing microdroplet evaporation. The apparatus includes alight source, a substrate to hold a microdroplet at a location, and afocusing module to focus the light source at an apex of themicrodroplet, wherein a beam waist of the focused light source has adiameter less than a diameter of the microdroplet.

In another example aspect, a method of evaporating a microdroplet isdisclosed. The method includes holding, by a substrate, a microdropletat a location, illuminating, by a light source, the microdroplet, andfocusing the light source at an apex of the microdroplet to cause adrying of the microdroplet, wherein a beam waist of the focused lightsource has a diameter less than a diameter of the microdroplet.

These, and other, aspects are described in greater details in thepresent document.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A depicts a laser and optics used to cause evaporation of adroplet, in accordance with some example embodiments;

FIG. 1B depicts a coated microarray with a pattern, in accordance withsome example embodiments;

FIG. 1C depicts an example of a microarray allowing the laser system tocontinuously deposit biomolecules from solution droplets on an XY plane,in accordance with some example embodiments;

FIG. 2A depicts an example of an image and line profile comparisonbetween fluorescent ssDNA (single-stranded DNA) molecule deposited on amicroarray by laser heating, in accordance with some exampleembodiments;

FIG. 2B depicts an example of an image and line profile comparisonbetween fluorescent ssDNA molecule deposited on a microarray with hotplate heating showing a coffee-ring pattern, in accordance with someexample embodiments;

FIG. 2C depicts an example table showing deposition pattern sizes andcapture ratio vs. drying method, in accordance with some exampleembodiments;

FIG. 3A depicts an example of a droplet undergoing differentialevaporation, in accordance with some example embodiments;

FIG. 3B depicts an example of a droplet undergoing evaporation due to ahot plate, in accordance with some example embodiments;

FIG. 3C depicts examples of contact radius time-evolution curves fordifferentially evaporated droplets and hot plate dried droplets, inaccordance with some example embodiments;

FIG. 4A depicts an example diagram showing laser heating at a dropletapex creating a temperature gradient and evaporation rate that ishighest at the droplet apex and lowest at the droplet base, inaccordance with some example embodiments;

FIG. 4B depicts an example diagram showing hot plate heating at thedroplet base creates a temperature gradient that is highest at thedroplet base and lowest at the droplet apex, in accordance with someexample embodiments;

FIG. 4C depicts an example diagram showing that due to the surface laserheating on the droplet apex, water evaporation flux is highest at thedroplet apex, in accordance with some example embodiments;

FIG. 4D depicts an example diagram showing that due to lower probabilityof water vapor reabsorption at droplet edge, water evaporation flux ishighest at the droplet edge, in accordance with some exampleembodiments;

FIG. 5A depicts an example of surface water vapor saturation pressureand droplet internal flow tracking by polystyrene beads fromlaser-induced differential evaporation for a 1 μL water droplet, inaccordance with some example embodiments;

FIG. 5B depicts an example of surface water vapor saturation pressureand droplet internal flow tracking by polystyrene beads fromlaser-induced differential evaporation for a 5 μL droplet, in accordancewith some example embodiments;

FIG. 5C depicts an example of polystyrene micro-bead movement trackingduring a 33 ms timeframe showing upward, centripetal fluidic flow, inaccordance with some example embodiments;

FIG. 6 depicts an example of model fitting of a drying patternanalytical solution to drying patterns of laser-induced differentialevaporation and hot plate heating, in accordance with some exampleembodiments;

FIG. 7 depicts an example of estimated droplet volume from parabolicprofile vs. exact droplet volume from spherical profile of dryingdroplets (n=3) during laser-induced differential evaporation;

FIG. 8 depicts {dot over (R)}/R and {dot over (θ)}/θ for of dryingdroplets (n=3) during laser-induced differential evaporation; and

FIG. 9 depicts an example table of observed beam size vs. distance tolens.

DETAILED DESCRIPTION

Disclosed is a general solution to remove the “coffee-ring effect”(hereinafter, coffee-ring effect), ubiquitous in the drying process ofaqueous droplets that causes many undesirable effects. Disclosed aremethods, devices and systems to cause differential evaporation ofmicro-droplets in a scalable fashion. In some embodiments, an attenuatedlaser is used to focus laser power to the center of an aqueous solutiondroplet, producing a differential evaporative flux profile that ispeaked at the droplet apex. The laser-induced differential evaporationdescribed herein is a significant improvement in the deposition ofwater-soluble molecules such as nucleic acids, proteins, inks, and othersmall molecules. The disclosed techniques enable new paradigms in liquidbiopsy combinational analysis, microarray fabrication, and ink-jetprinting.

Reversing the coffee-ring effect which is ubiquitous in the dryingprocess of aqueous droplets and causing many adverse effects enables newparadigms in liquid biopsy combinational analysis, microarrayfabrication, and ink-jet printing. Part of this patent documentdescribes the role of differential evaporation in the transition of adeposition pattern from a coffee-ring to a center peak pattern. Tocreate the preferred differential evaporation for micro-droplets in ascalable fashion, a 10 W CO₂ laser may be used to focus 40 mW laserpower to the center of the aqueous solution droplet, producing adifferential evaporative flux profile peaked at the droplet apex. Thisdifferential evaporation flux profile then leads to an inward, radialflow that reverses the coffee-ring effect. This laser-induceddifferential evaporation method may be used on a fluorescent DNAsolution to produce a shift in the deposition pattern, on a polystyrenebeads solution to track internal fluidic flow, and on pure water todemonstrate the differential evaporation.

Reversing the coffee-ring effect in aqueous droplets enables controlled,microscale deposition of water-soluble molecules, which is important formany applications including liquid biopsy combinational analysis,microarray fabrication, and ink-jet printing. The disclosed methodreverses the coffee-ring effect without using any surfactant oradditive, thus providing minimal interference on downstream processes.As disclosed, a CO₂ laser beam may be focused to the center of aqueoussolution droplet to generate differential evaporation, which leads to aninward, radial flow within the droplet and reverses the coffee-ringeffect. Since the laser-induced differential evaporation process useslow power compared to the output of a compact CO₂ laser, the approachcan be scaled up to support high throughput applications.

The drying of a droplet of water carrying colloidal particles naturallygives rise to non-homogenous deposition pattern, with most of theparticles migrating to the edge of the droplet, forming the well-knowncoffee-ring deposition pattern. This non-uniform deposition has posedchallenges in ink-jet printing, DNA/RNA and protein microarraymanufacturing, and most recently in combinational liquid biopsy analysismethods such as fluorescent microarray, infrared spectroscopy, and Ramanspectroscopy.

The cause of the coffee-ring effect is an outward, radial flow initiatedby a weakly pinned contact line and maximum evaporation flux at the edgeof the droplet. Thermally-induced Marangoni effect is effective increating a re-circulation flow to reverse the coffee-ring effect, and toreduce the deposition pattern size of solution droplets with volatilesolvents. However, when applied to aqueous droplets, thethermally-induced re-circulation flow is suppressed and the coffee-ringeffect dominates the Marangoni effect. Electro-wetting is also effectivein reducing the deposition pattern size of drying aqueous droplet butrequires relatively high concentration (10 mM) of LiCl additive toadjust the solution's conductivity. To recover signals of diluted (100nM to 100 aM) analyte from the dried LiCl solid crystal poses yetanother technical challenge. To improve aqueous sample analysis from adrying droplet, one or more of the following are satisfied: reversing ofcoffee-ring effect, reduction in deposition pattern size, and minimal orpreferably no additive to the solution.

The coffee-ring effect can be removed effectively provided there is amethod to produce an environment that the evaporation rate at the apexof the droplet is much faster than the evaporation rate on surfaceselsewhere. The strong evaporation rate at the central region of thedroplet will provide the needed driving force for the flow thatsuppresses the coffee-ring effect. Given the small size of the dropletsin most applications, a CO₂ laser may be used to cause laser-induceddifferential evaporation. Water molecules have strong absorption(greater than 3000 cm-1) at 10.6 μm wavelength produced by a CO₂ laser.As such, a CO₂ laser will generate a differential evaporative fluxprofile on a water surface. Using the disclosed techniques, thelaser-induced differential evaporation method has been used to reducethe deposition pattern size of aqueous solution droplets from 1.5 mm toa 100 μm spot without the addition of ionic salts or surfactants. Thedisclosed system and techniques are low-power (40 mW) and can be scaledup with parallel optical paths in an array format. In this way, manydroplets can be evaporated at the same time (e.g., within 1 to 10nanoseconds, or within one microsecond of each other). During dropletevaporation, the CO₂ laser beam creates maximum evaporative flux at theapex of the droplet, causing an inward, radial flow that dominates andcounteracts the outward, radial flow of the coffee-ring effect. Theinward flow occurs concurrently with contact line de-pinning andultimately leads to the peak deposition patterns.

FIGS. 1A-1C depict an example of a system for differential evaporation,in accordance with some example embodiments. FIG. 1A depicts a lightsource 110 which can be a carbon dioxide laser producing beam 115 thatis collimated through a beam expander (not shown), reflected at a 45°angle 3 times at 120, and attenuated 75% (attenuator not shown) toachieve top-down exposure of the plano-convex lens 130. Through theplano-convex lens 130, the laser beam converges to a focal point 140.Using an alignment microscope 145 and a XYZ linear stage 150, a sampledroplet 160 is centered to the laser beam's focal point and a highevaporation flux occurs at the apex of the droplet. FIG. 1B depicts acytop coated microarray with 4×4 100 μm patterns with 2 mm spacing. FIG.1C depicts a microarray allowing the laser system to continuouslydeposit biomolecules from solution droplets on an XY plane.

The disclosed CO₂ laser system generates maximal evaporation fluxdifference on the droplet surface. To realize such effect, an examplesystem focuses the CO₂ laser beam to a 28.9 μm spot on the glasssubstrate, which is the basal surface of the droplets. In theopto-mechanical and imaging setup 100 shown in FIG. 1A, the light source110 can be a 10 W CO₂ laser (e.g., Universal Laser Systems ULR-10) withemission wavelength of 10.6 μm, beam size of 4 mm, and divergence of 5milliradians (mrad). The laser power is modulated at a frequency of 20kHz and duty cycle ranging from 0-100%. To achieve the desired size ofthe beam spot, the laser beam enters a 10× beam expander including apair of plano-concave (with a focal length of 50 mm, for example and adiameter of 1″) and plano-convex lens (with a focal length 500 mm, forexample, and a of diameter 1″). The lenses may be made from zincselenide (ZeSe) and have a broadband (7-12 μm) antireflective coating.The expanded laser beam with diameter of 40 mm is reflected three timesat a 45° angle at 120 using gold-coated mirrors to allow the beamdirection to change from parallel to perpendicular to the sample stage.Since differential evaporation requires much lower laser power (e.g., 40mW) that the laser produces (e.g., 10 W), a piece of 22 mm glasscircular coverslip can be fixed to the center of one of the gold-coatedmirrors to achieve 75% attenuation. Other techniques can also be used tocause the desired attenuation. The laser beam then passes through aplano-convex 130 lens (with focal length 50 mm, for example, and adiameter 1″) made using ZeSe that converges into a focal point. Theposition of the laser focal point is recorded using an alignmentmicroscope 145 (e.g., Dino-Lite AM411T) mounted at a 30-40° angle fromthe sample stage.

To define the sites for sample enrichment and/or reactions(hybridization or binding) in an array format, the glass substrate maybe coated with amorphous fluorocarbon polymer such as cytop with 100μm-diameter SiO₂ wells. Since a cytop-coated surface is hydrophobic(100° in contact angle, 4.5 μm in thickness), the cytop microarraydevice surface is hydrophobic except the uncovered, 100 μm-diameter SiO₂wells (43° in contact angle), as shown in FIG. 1B. The cytop coatingencourages water droplets to de-pin from their surfaces while thehydrophilic well anchors the droplets onto the pattern throughout thedrying process. A cytop microarray device can have dimensions of 8 mm by8 mm and can have 4×4 100 μm well patterns spaced 2 mm apart. Themicroarray device may have other dimensions, well patterns, and/orspacings as well. During experimental operation, sample droplets may bespotted onto patterns using, for example, a syringe-pump dropletspotting system. To start the differential evaporation process, thelaser is turned on at 40 mW and the laser beam is focused onto thespotted droplet (see, for example, FIG. 1C). As the sample dropletdries, the laser power is reduced sequentially to be approximatelyproportional to the 3rd power of the receding radius of the droplet.

To demonstrate the effect of laser-induced differential evaporation onbiomolecule deposition, 1 μL droplets have been dried that contain 10 nMfluorescent ssDNA (single stranded FAM labeled DNA, with a length, forexample, of 24 nts) using both laser-induced differential evaporationmethod and a hot plate setup. The hot plate method accelerated solutiondroplet evaporation by heating up droplets from the bottom liquid-solidcontact.

FIGS. 2A-2B depict examples of image and line profile comparisonsbetween fluorescent ssDNA (single-stranded DNA) molecule (FAM labeled,24 base pairs in length) deposited on a cytop microarray by laserheating and hot plate heating (50 C). FIG. 2A depicts an example ofdifferential evaporation by laser heating of DNA molecule depositionpatterns confined within the 100 μm patterns. In some exampleembodiments, the average time for 1 μL of 10 nM solution droplet to dryis ˜90 s. FIG. 2B depicts example hot pate heating results with atypical coffee-ring pattern, in which the DNA molecule dries in a ringstructure (˜1.5 mm) outside of the 100 μm patterns. In some exampleembodiments, the average time for 1 μL of 10 nM solution droplet to dryis ˜480 s for a 50 C hot plate.

As noted above, FIGS. 2A-2B show fluorescent DNA deposition patterns ona cytop microarray device resulting from both the laser-induced methodand the hot plate heating (50 C). The deposition patterns were imagedusing an enclosed fluorescent microscope. For the differentialevaporation method, fluorescent image and line profile analysis showsinsignificant fluorescent signal outside of the 100 μm microarraypattern (FIG. 2A), indicating minimal liquid pinning event during dryingand an absence of droplet edge-deposition. For the hot plate method,fluorescent image and line profile analysis show prominent fluorescentsignal in a ring-structure with a diameter ˜1.5 mm, indicatingsignificant liquid pinning event during drying and dropletedge-deposition, the coffee-ring effect (FIG. 2B). In terms of overallevaporation rate, the differential evaporation method requires ˜90 secto dry 1 μL of the 10 nM solution droplet, and the hot plate heatingrequires ˜480 sec for complete evaporation. With a 16×16 ZnSe microlensarray with 1.5 mm pitch, the method can be extended to dry 256 droplets,or more, simultaneously with the compact CO₂ laser.

Despite sharing fluidic properties, bottom substrate surface materialand geometry, and ambient environment conditions with the hot plateheating method, the differential evaporation method shows no coffee-ringdeposition. FIG. 2C (Table 1) shows example results (n=6) of depositionpattern size and capture ratio from both differential evaporation andhot plate methods (1 μL of 10 nM fluorescent ssDNA). Pattern sizequantifies the spread of the deposition pattern, and capture ratiomeasures the relative amount of fluorescent signal detected in the 100μm pattern region compared to the total signal of the imaged surface.For the differential evaporation method, am example of repeatingdeposition testing yields an average drying pattern size of 100.8 μm,size standard deviation of 1.28 μm, and an average capture ratio of73.21%, indicating that most fluorescent DNA molecules were depositedwithin the pattern. The capture ratio being less than 100% may be due todevice background auto-fluorescence, background light scattering, ornon-specific DNA surface adsorption. Nevertheless, in the case ofcoffee-ring deposition by hot plate heating, the average capture ratiois 0.31%. A low capture ratio, combined with an average pattern size of1504 μm, indicates that most fluorescent DNA are deposited outside ofthe microarray 100 μm pattern. Both qualitative and quantitativeanalysis indicates differential evaporation causes a shift in the modeof deposition from the typical coffee-ring deposition.

To establish the physical connection between differential evaporationand the mode of biomolecule deposition, the mode of droplet evaporationduring differential evaporation is determined. The mode of dropletevaporation characterizes the history of a droplet contact line positionand contact angle. While there are two pure droplet evaporation modes,one with constant contact radius (CCR) and the another one with constantcontact area (CCA), most aqueous droplet evaporation undergoes both puremodes as well as a mixture of the two modes where both contact area andcontact radius are variable. The transition from the CCA mode to the CCRmode is related to pinning of the triple phase contact line. FIGS. 3A-3Bshows the droplet evaporation mode during both differential evaporationand hot plate heating (50 C) on cytop microarray device (1 μL, of 10 nMfluorescent ssDNA). FIG. 3A depicts an example including 1 μL of 10 nMssDNA droplet undergoing differential evaporation. The droplet shape isfitted at each time to a corresponding parabolic curve 310A-310E. Astime progresses (progressing from 310A-310E), the droplet volume andcontact radius shrinks due to water evaporation at droplet apex.Micro-droplets condense around the heated droplet due to water vapordiffusion from active evaporation and temperature gradient of therelatively cooled bottom surface. FIG. 3B depicts an example including 1μL of 10 nM ssDNA droplet heated by a 50 C hot plate. The droplet shapesare fitted at each time to a corresponding parabolic curve 320A-320E. Astime progresses (progressing from 320A-320E), the droplet volume shrinksdue to water evaporation from bottom hot plate heating. No surfacecondensation is observed since the solid surface is higher intemperature. FIG. 3C depicts an examples of contact radiustime-evolution curves. Contact radius for differentially evaporateddroplet continues to shrink with time, while hot plate dried dropletsare pinned at their original position.

Droplet evaporation recordings reveal two differences between the twoevaporation processes. One difference is the surface condensation on thecytop microarray during differential evaporation, forming micro-dropletsaround the original droplet, while hot plate heating is free of surfacecondensation (FIG. 3A). In the hot plate heating, the device surfacetemperature is elevated to 50 C relative to the ambient temperature (˜20C) and thus water vapor rapidly leaves the surface and diffuses into theambient (FIG. 3B). On the other hand, during differential evaporation,the substrate surface is at ambient temperature, which may causecondensation of the hot vapor oversaturating the surrounding region.Another difference is contact line movement. During differentialevaporation (FIG. 3C), the droplet contact line continues to de-pin fromthe cytop surface and moves toward the center, while hotplate heatingresults in a pinned contact line throughout droplet evaporation. Thecontact radius time-evolution curve shows a time-independent, constantcontact radius of 750 μm for hot plate heating, matching the CCR mode ofevaporation. In contrast, laser-induced differential evaporation resultsin a receding contact radius with time progression. Moreover, during the0.4 to 0.8 T/T₀ time regime, the normalized contact radius receding rateis linear (i.e. {dot over (R)}/R˜Ct), showing the characteristic of theCCA mode of evaporation. While the normalized contact angle rate fordifferential evaporation (i.e., {dot over (θ)}/θ) declines with time andessentially approaches zero, again showing the characteristic of the CCAmode of evaporation. Comparing between contact radius and contact angledata for the two processes, a shift from CCR to CCA evaporation mode isobserved for the laser-induced differential evaporation method.

The cause behind contact line pinning is a weakly pinned contact lineand the suppression of thermally-induced flow, which leads to anoutward, radial flow causing the coffee-ring effect. FIG. 4 shows liquidtemperature isotherm, water evaporation flux distribution, and internalfluidic flow diagrams for both laser-induced differential evaporationand hot plate heating methods. The liquid isotherm diagram shows laserinduced evaporation creates a temperature profile that is highest atdroplet apex and lowest at droplet base (FIG. 4A). In contrast, hotplateheating creates a temperature profile with the highest temperature atdroplet base (FIG. 4B).

FIGS. 4A-4D depict thermal and water evaporation flux response to laserheating. FIG. 4A depicts laser heating at the droplet apex creating atemperature gradient and evaporation rate that is highest at dropletapex and lowest at droplet base. FIG. 4B depicts hot plate heating atthe droplet base creating a temperature gradient that is highest atdroplet base and lowest at droplet apex. FIG. 4C depicts, waterevaporation flux is highest at the droplet apex due to the surface laserheating on the droplet apex. An internal flow develops to replenish thelost water volume at the apex, producing an inward, radial flow thatcarries colloidal particles to the droplet center and reverses thecoffee-ring effect. FIG. 4D depicts water evaporation flux is highest atthe droplet edge due to lower probability of water vapor reabsorption atdroplet edge. Governed by the mass transport equation, an internal flowmust be supplied to replenish the water loss at the edge. As a result,an outward, radial flow carries colloidal particles to the droplet edge,causing the coffee-ring effect.

To understand how laser heating leads to differential evaporation at thedroplet's surface, the relation between water evaporation flux J* andinterface quantities via the Hertz-Knudsen expression from kinetictheory of gas,

${J^{*} = {\alpha {\sqrt{\frac{M}{2\pi \; \overset{\_}{R}T_{sat}^{*}}}\left\lbrack {{p_{sat}^{*}\left( T_{i}^{*} \right)} - p_{v}^{*}} \right\rbrack}}},$

where p_(sat)*(T_(i)*) is the saturation pressure at the interfacetemperature T_(i)*, T_(sat)* is the saturation temperature, p_(v)* isthe vapor pressure just beyond the interface, and R is the universal gasconstant. The parameters α and M are the accommodation coefficient(measure of liquid volatility) and the molecular mass of vaporrespectively. For most practical cases, the coefficients α, M, R, andT_(sat)* can be considered nearly constant throughout the process ofdroplet evaporation. For laser induced differential evaporation in FIG.4C, J* reaches maximum at the droplet apex because surface temperaturemaximum yields maximum [p_(sat)*(T_(i)*)−p_(v)*]. Moreover, p_(v)* ismuch lower than p_(sat)*(T_(i)*) above the droplet apex because thelaser beam also heats up the water vapor just beyond the droplet apexsurface to drive the vapor away. The combined effect of elevated surfacesaturation pressure and removal of external water vapor gives rise tothe laser-induced differential evaporation phenomenon. Based on theequation of mass transport, the strong water evaporation flux at thedroplet apex leads to an inward, radial flow toward the droplet center,carrying colloidal particles toward the center of droplet. For the hotplate heating analysis in FIG. 4D, local evaporation rate diverges andreaches maximum at the droplet edge because of lower probability ofwater vapor reabsorption at droplet edge. Therefore, an outward, radialflow develops, carrying the colloidal particles to the droplet edge toproduce the coffee-ring effect.

To support the forgoing explanation, the local saturation pressure anddroplet internal flow direction may be experimentally verified. Sincethere is no direct method to assess the local saturation pressuresurrounding the droplet, the droplet surface temperature is recorded toextrapolate the saturation pressure at the surface. For example, thedroplet surface temperature is imaged using a microbolometer (FLIRA655sc) at a resolution of 640×480 pixels and at 25 μm spatialresolution. To measure the internal fluid flow, polystyrene micro-beadsare added to the solution to track their motions from recorded images.

FIGS. 5A-5C depict surface water vapor saturation pressure and dropletinternal flow tracking by polystyrene beads from laser-induceddifferential evaporation. FIG. 5A depicts laser-induced differentialevaporation (40 mW) on 1 μL water droplet yielding a saturation pressureof 234 mmHg on the droplet apex surface. FIG. 5B depicts laser-induceddifferential evaporation (40 mW) on a 5 μL droplet yielding a saturationpressure of 49.7 mmHg on the droplet apex surface. Oversaturated spotsin (A) and (B) are artifacts caused by reflections of the IR laser. FIG.5C depicts polystyrene micro-beads (3.3 μm in diameter, 800 beads/μL inwater) movement tracking during a 33 ms timeframe shows upward,centripetal fluidic flow with a mean velocity of 11.2 mm/s toward thedroplet apex in the 1 μL droplet.

As noted above, FIGS. 5A-5D shows surface water vapor saturationpressure and polystyrene bead movement during laser-induced differentialevaporation. Two volumes of water droplet (5 μL and 1 μL) are imaged todemonstrate the direct effect of laser beam size on surface water vaporsaturation pressure (FIGS. 5A-B). The laser beam diameter is ˜100 μm and˜60 μm respectively on 5 μL (˜2 mm of liquid height) and 1 μL (˜0.85 mmof liquid height) water droplet apex. Under the same laser power of 40mW, the 100 μm and 60 μm laser beam diameters correspond to saturationpressure of 234 mmHg and 49.7 mmHg on droplet apex. This observationcorrelates to different progression during the differential evaporationprocess. As the droplet size reduces, surface saturation pressureincreases at droplet apex. Tracking the internal microfluidic flow, thepolystyrene bead solution droplets are imaged at 30 frames per second(FIG. 5C). The polystyrene beads are 3.3 μm in diameter and suspended inwater at the concentration of 800 beads/μL. The bead positions wereanalyzed and overlaid onto the original images using a custom MATLABprogram. During a timeframe of 33 ms, beads starting at the centricpositions (#1,4,6,7) moves toward the apex with a mean velocity of 13.3mm/s while the edge positions (#2,3,5,8) have a mean velocity of 9.18mm/s, suggesting strongest flow at droplet center. Overall, polystyrenemicro-beads migrate toward the droplet apex at a mean velocity of 11.2mm/s, in agreement with the proposed coffee-ring reversion flow profile.

To elucidate experimental results of laser-induced differentialevaporation and to support the qualitative arguments how the proposedmethod can remove the coffee-ring effect, a physical model to connectdifferential evaporation to coffee-ring effect reversion is describedbelow. The following two conditions are approximately satisfied:

$\begin{matrix}{{{{Droplet}\mspace{14mu} {height}\mspace{14mu} {profile}\text{:}\mspace{14mu} {h\left( {r,t} \right)}} = {{H(t)}\left\lbrack {1 - \frac{r^{2}}{R^{2}(t)}} \right\rbrack}},} & {{Equation}\mspace{14mu} (1)} \\{{{Overall}\mspace{14mu} {evaporation}\mspace{14mu} {rate}\text{:}\mspace{14mu} \overset{.}{V}} = {{\overset{.}{V_{o}}\left( \frac{R(t)}{R_{o}} \right)}.}} & {{Equation}\mspace{14mu} (2)}\end{matrix}$

In Equation (1), the droplet height profile h(r,t) follows a paraboliccurve described by its center height H(t) and contact radius R(t) attime t. Equation (2) describes how the evaporation rate {dot over (V)}is related to the initial evaporation rate {dot over (V)}_(o) with R(t)and R_(o) being the droplet contact radius at time t and time 0. FIG.3A-B shows the parabolic curve fit to droplet shape during bothlaser-induced evaporation and hot plate drying. After initial 30% ofdrying time progression, parabolic curves fit droplet shapes, showingEquation (1) is accurate for T/T₀>0.3. During the laser-heating process,after initial 30% of drying time progression, the evaporation rate isproportional to contact radius and independent of contact angle, andthus Equation (2) is accurate for T/T₀>0.3.

The solvent mass conservation equation can be expressed as:

$\begin{matrix}{{{\frac{d}{dt}{\int_{0}^{r}{{dr}^{\; \prime}2\pi \; r^{\prime}{h\left( {r^{\prime},t} \right)}}}} = {{{- 2}\pi \; {{rv}\left( {r,t} \right)}{h\left( {r,t} \right)}} - {\int_{0}^{r}{{dr}^{\; \prime}2\pi \; r^{\prime}{J\left( {r^{\prime},t} \right)}}}}},} & {{Equation}\mspace{14mu} (3)}\end{matrix}$

where v(r,t) denotes the height-average fluid velocity and J(r,t)denotes the local solvent evaporation rate per unit area (Vol/s-area).The sign of the velocity is positive for fluid leaving the center andnegative towards the center. The following profile for J(r,t) may beused:

$\begin{matrix}{{J\left( {r,t} \right)} = {- {\frac{\overset{.}{V_{o}}}{{\pi R}_{o}{R(t)}}.}}} & {{Equation}\mspace{14mu} (4)}\end{matrix}$

To describe laser-induced differential evaporation, J(r,t) in Equation(4) may be modified as J(r,t)=J_(i)+J_(d), where J_(i) is theevaporation rate due to isothermal evaporation because of laser heating,and J_(d) is the differential evaporation by the focused laser beam.From the measured temperature profile of the droplet under the focusedCO₂ laser beam and the fact that laser-induced evaporation dries thedroplet 10 times faster than the uniformly heated droplet to 50 C by ahot plate, J_(d)>>J_(i).

$\begin{matrix}{{{J\left( {r,t} \right)} \sim J_{i} \cong {0\mspace{14mu} {if}\mspace{14mu} r} > a}{{{J\left( {r,t} \right)} \sim J_{d}} = {{\frac{P}{a^{2}}\mspace{14mu} {if}\mspace{14mu} r} < {a.}}}} & {{Equation}\mspace{14mu} (5)}\end{matrix}$

In this expression, laser-induced evaporation rate P(t) and laserexposure radius a. P(t) has the unit of vol/s and is proportional to thedroplet surface temperature in the laser irradiated area and to thelaser power. Substituting Equation (5) into Equation (3) and expandingeach individual term, the following relation may be obtained:

$\begin{matrix}{{{2\pi \; {{rH}(t)}v} = {{- {\frac{\pi \; r^{2}\overset{.}{H}}{2}\left\lbrack {1 + \frac{1}{\left( {1 - \frac{r^{2}}{R^{2}}} \right)}} \right\rbrack}} + {\pi \; r^{2}H{\frac{\overset{.}{R}}{R}\left\lbrack {1 - \frac{1}{\left( {1 - \frac{r^{2}}{R^{2}}} \right)}} \right\rbrack}} - \frac{P}{\left\lbrack {1 - \frac{r^{2}}{R^{2}}} \right\rbrack}}},} & {{Equation}\mspace{14mu} (6)}\end{matrix}$

where {dot over (H)} is the time derivative of droplet center height.Utilizing Onsager's principle 22,

${\frac{\partial\left( {\Phi + \overset{.}{F}} \right)}{\partial\overset{.}{R}} = 0},$

Equations (1), (2), and (6), and the assumption that the droplet volumedecreases much faster than the equilibration of the contact angle(K_(ev)>>1), the rate of contact radius movement {dot over (R)} can beapproximated as:

$\begin{matrix}{{\overset{.}{R} \sim {\frac{R\overset{.}{V}}{4\left( {1 + k_{cl}} \right)V} + {\frac{RP}{4{V\left( {1 + k_{cl}} \right)}}\left( {1 + \frac{1}{C}} \right)}}},} & {{Equation}\mspace{14mu} (7)}\end{matrix}$

where k_(cl) denotes the ratio of contact line friction to hydrodynamicfriction and C=[−ln(∈)−1]. ∈ is a small number introduced to avoidsingularity and is defined as

$\epsilon = {\lim\limits_{r->{{({1 - 10^{- 6}})}R}}{\left( {1 - \frac{r^{2}}{R^{2}}} \right).}}$

In the analysis, it may be assumed that k_(cl) is a time-independentmaterial parameter that is determined by the droplet and the substrate.To analyze the coffee-ring effect, where and when the solutesprecipitate during the droplet evaporation process are found. It may beassumed that the solute moves at the same velocity as the fluid insidedroplet before precipitation. {tilde over (r)}(r_(o), t) may be definedas the height-averaged position of a solute at time t with their initialradial position at r_(o), and ({tilde over ({dot over (r)})}) as itstime derivative representing the speed of movement of the solute as thedroplet evaporates. Based on the above assumption for k_(cl) and fromEquation (7), the following relation between the solute speed, soluteposition, and the state of the droplet under laser-induced evaporationmay be obtained:

$\begin{matrix}{{{\frac{\left( \overset{.}{\overset{\sim}{r}} \right)}{\overset{\sim}{r}}\left( {r_{0},t} \right)} = {{{{- k_{cl}}\frac{\overset{.}{R}}{R}} - {{\frac{P}{2\pi \; H}\left\lbrack {\frac{1}{{\overset{\sim}{r}}^{2}} - {\frac{1}{R^{2}}\left( {1 + \frac{1}{C}} \right)}} \right\rbrack}\mspace{14mu} {for}\mspace{14mu} \overset{\sim}{r}}} > a}},} & {{Equation}\mspace{14mu} (8)}\end{matrix}$

The negative sign in front of the

$\frac{\overset{.}{R}}{R}$

term means that the solute with an initial position r_(o) moves in theopposite direction to the droplet radius without laser inducedevaporation (i.e. P=0). At the time t_(d) when {tilde over (r)}==R, thesolute precipitates at the edge of the droplet, revealing thecoffee-ring effect. The greater is the coefficient k_(cl) that isrelated to liquid viscosity and contact line friction, the more seriousthe coffee-ring effect becomes. However, when the laser-inducedevaporation is set at an appropriate level (to be determined next), andprovided the solute at position r_(o) initially (t=0) precipitates att=t_(d) at droplet edge ({tilde over (r)}(r_(o),t_(d))=R), the velocityof solute ({tilde over ({dot over (r)})}) (r_(o),t_(d)) can be in thesame direction as {dot over (R)} but at a higher magnitude, yielding acondition that contradicts the presumption that the solute precipitatesat the edge of the droplet. Therefore, when the laser induceddifferential evaporation rate reaches an appropriate level, solute withits initial position r_(o) will not precipitate at the edge of thedroplet, thus removing the coffee-ring effect. A more detailedquantitative analysis is discussed next.

A comparison may be made between the physical model and experimentalresults. Since Equation (8) is a function of the laser-inducedevaporation rate P(t), P(t) can be controlled to obtain the desiredprecipitation pattern of solute. A practical approach is to define amathematical expression for the precipitation profile and find therequired laser induced evaporation rate such that the resultingprecipitation profile is bounded by the mathematical expression. FromEquation (8), it can be found that without laser-induced evaporation,the position of the solute is related to the droplet radius as:

${{\overset{\sim}{r}\left( {r_{0},t} \right)} = {r_{0}\left( \frac{R}{R_{0}} \right)}^{- k_{cl}}},$

which will lead to the coffee-ring pattern as explained previously. Tocounter the coffee-ring effect, we introduce a parameter G>0 to alterthe relation into (9) and use the laser power to control the value of G:

$\begin{matrix}{{{\overset{\sim}{r}\left( {r_{0},t} \right)} = {r_{0}\left( \frac{R}{R_{0}} \right)}^{- {\lbrack{k_{cl} - G}\rbrack}}},} & {{Equation}\mspace{14mu} (9)}\end{matrix}$

To overcome the coffee-ring effect, G needs to be of sufficient strengthsuch that G>k_(cl)+1.27. From Equations (8) and (9) and the criterionfor G, we show that P(t) needs to satisfy the following criterion:

$\begin{matrix}{{{P(t)} > {{- {GR}}{\overset{.}{R}\left( {2\pi \; H} \right)}}} = {\frac{G\; \theta}{2}{{{\frac{d}{dt}\left( \frac{2\pi \; R^{3}}{3} \right)}}.}}} & {{Equation}\mspace{14mu} (10)}\end{matrix}$

To obtain the last expression of Equation (10), the relation that thecontact angle

$\theta \sim \frac{2H}{R}$

is time independent for the CCA mode is applied. The term

${\frac{d}{dt}\left( \frac{2\pi \; R^{3}}{3} \right)}$

is the rate of change of the volume of “hypothetical half dome” of thedroplet even though the actual shape of the evaporation droplet is notsemi-spherical.

The result suggests that by controlling the laser power according toEquation (10), the solute precipitation behaviors will be within thebound of Equation (9). During our laser-heating process, laser power wasgradually decreased from 40 to 0 mW over the 90 second drying period, assuggested by Equation (10).

Assuming the amount of solute initially present between ro and ro+dro islater on precipitated between {tilde over (r)}=and {tilde over(r)}+d{tilde over (r)}, we then have the relation2πØ_(o)h(r_(o))r_(o)dr_(o)=2πu({tilde over (r)}){tilde over (r)}d{tildeover (r)}, which can be written as:

$\begin{matrix}{{u\left( \overset{\sim}{r} \right)} = {Ø_{o}{h\left( r_{o} \right)}\frac{r_{0}}{\overset{\sim}{r}}{\left( \frac{d\overset{\sim}{r}}{{dr}_{o}} \right)^{- 1}.}}} & {{Equation}\mspace{14mu} (11)}\end{matrix}$

where Ø_(o) is the initial solute concentration and u({tilde over (r)})is the drying pattern deposit density.

The solute precipitation condition {tilde over(r)}(r_(o),t_(d))=R(t_(d)) at time t_(d), Equation (9) gives rise to thefollowing relation:

$\begin{matrix}{{\overset{\sim}{r}\left( {r_{o},t_{d}} \right)} = {{R\left( t_{d} \right)} = {r_{o}^{\frac{1}{1 + k_{cl} - G}}{R_{o}^{\frac{k_{cl} - G}{1 + k_{cl} - G}}.}}}} & {{Equation}\mspace{14mu} (12)}\end{matrix}$

Substituting Equation (12) into Equation (11), the drying patterndeposit density may be determined as:

$\begin{matrix}{{{u\left( \overset{\sim}{r} \right)} = {Ø_{o}{H_{o}\left( {1 + k_{cl} - G} \right)}{\left( \frac{\overset{\sim}{r}}{R_{o}} \right)^{2{\lbrack{k_{cl} - G}\rbrack}}\left\lbrack {1 - \left( \frac{\overset{\sim}{r}}{R_{o}} \right)^{2{({1 + k_{cl} - G})}}} \right\rbrack}}},} & {{Equation}\mspace{14mu} (13)}\end{matrix}$

where H_(o) is the initial droplet height at t=0. In FIG. 6,

$\frac{u\left( \overset{\sim}{r} \right)}{Ø_{o}H_{o}}$

is normalized to its maximum value and plotted against

$\frac{\overset{\sim}{r}}{R_{o}}$

to show changes of drying pattern deposit density from center to edge ofthe initial droplet.

FIG. 6 depicts an example of fitting a drying pattern analyticalsolution to drying patterns of laser-induced differential evaporationand hot plate heating. Solute densities are normalized to theirrespective maximum values. The droplet volume is assumed to decreasemuch faster than the equilibration of the contact angle (K_(ev)>>1).k_(cl) is a time-independent material parameter that is determined bythe droplet and the substrate. G is related to laser-induced evaporationrate P(t) by Equation (10). For all cases, material parameter k_(cl)=3.For G=0 (purple curve), the analytical solution closely matches to thedrying pattern of hot plate heating, a characteristic coffee-ring. ForG=4.5 (orange curve), the analytical solution closely matches to thedrying pattern of laser-induced differential evaporation, acharacteristic center peak.

Because DNA concentration was enriched from its initial concentration of10 nM and the heated evaporations (laser and hot plate) occurred fasterthan natural drying, K>>1. To fit the drying pattern of hot-platehitting, the material parameter k_(cl)=3 may be chosen. The result showsthe coffee-ring effect. By increasing the laser-induced evaporation rateto increase the value of C, the deposit peak moves toward the dropletcenter. At G=4.5, the drying pattern shape shifts from coffee-ring tocenter peak pattern, in agreement with the observed drying pattern oflaser-induced differential evaporation. Therefore, experimental resultsabout laser power dependence on the droplet size are agreed with apossible model.

In light of the above features, the disclosed technology can beimplemented to use a low power, scalable CO₂ laser setup to producedifferential evaporation over a droplet to mitigate the coffee-ringeffect without any surfactants or additives. As the droplet dries, thesolutes precipitate within a predefined area at the center of thedroplet. The disclosed technology allows enrichment and focuseddeposition of water-soluble molecules, and has potential tosubstantially advance the technologies in combinational liquid biopsyanalysis, ink-jet printing, and microarray fabrication.

An example droplet spotting system setup is now described. FluorescentDNA molecules may be diluted in Milli-Q water and spotted using theintegrated syringe pump system. The system includes a programmablesyringe pump (NE-1000, New Era Pump System) mounted with a 1 mL plasticsyringe (Tuberculin Syringe, Becton Dickinson). The syringe tip may beconnected to a #27 gauge, stainless metal tip dispensing needle (I.D.210 μm) via plastic mount and then extended with a 30 cm segment ofTygon tubing before interfacing with a #27 gauge stainless metal tipremoved of its plastic stage. For precise displacement control,stainless metal tip at the end of tubing was fixed onto a probe holderintegrated to an XYZ linear stage.

An example cytop microarray fabrication is now described. A cytopmicroarray may be patterned with cytop polymer (e.g., Asahi Glass Co.,Japan) on 75×50×1 mm glass slides (Thermo Fisher Scientific, USA).Before cytop coating, the glass slide was solvent-cleaned and dried.Cytop polymer type A, containing carboxyl end functional group, was usedfor coating. 0.05% of (3-aminopropyl) triethoxysilane (e.g., SigmaAldrich, USA) in ethanol/water (95/5) mixture were spin-coated on glassto promote cytop adhesion. A 4.5 μm thick cytop polymer layer was formedon glass by spin-coating 9% cytop type A solution at 800 rpm for 30 sand cured at standard cytop curing condition. To promote photoresistadhesion, cytop surface was oxygen plasma treated in a microwave plasmasystem (PS100, PVA Tepla) at 2.45 GHz frequency, gas flow rate of 120sccm, and power of 200 W for 60 s. Using conventional photolithographymethod, a 5 μm thick negative tone photoresist NR-9 6000PY (Futurrex,USA) was spin-coated onto the cytop-coated glass and patterned with 100μm circular opening to cytop surface. 100 μm well patterns onphotoresist were transferred onto the cytop coating by oxygen plasmaetching (Plasmalab 80 plus system, Oxford Instruments) the exposed cytopsurface. After complete etching of cytop surface, the remainingphotoresist was removed by immersion in resist remover (RR41, Futurrex).

Deposition pattern imaging and analysis. Fluorescent DNA drying patternon cytop-coated microarray devices may be imaged using an enclosedinverse fluorescent microscope (e.g., BZ-9000, Keyence Corporation) ateither 5× or 20× magnification depending on the pattern size. Thesamples were excited by a mercury lamp through a single-band bandpassfilter (472.5/30 nm), and the emission light was filtered by anothersingle-band bandpass filter (520/35 nm). Both pattern size and captureratio analysis were implemented using the ImageJ software. To estimatefor the pattern size, ferret diameter of the best-fitting ellipticalshapes to the fluorescent DNA pattern was calculated and used. Tocalculate the “capture ratio” for any given pattern, the integratedfluorescent intensity within the 100 μm microarray pattern was dividedby the total integrated fluorescent intensity of the image. Accordingly,the apparatus may achieve a beam waist of laser (e.g., CO2 laser) to bebetween 25 and 35 microns.

The laser beam is further characterized below. The diffraction limitedbeam spot, D_(fp), can be calculated from the equation

$D_{fp} = \frac{2\mspace{11mu} \lambda}{\pi \times {N.A.}}$

where λ is the laser wavelength and N.A. is the numerical aperture of anoptical setup. In our system

${{N.A.} = \frac{D_{beam}}{2\mspace{14mu} f}},$

where D_(beam) is the laser beam size entering the final plano-convexlens and f is the focal length. Given our laser wavelength, beam size,and focal length being λ=10.6 μm, D_(beam)=1″, and f=50 mm, thediffraction limit of a focused CO2 laser beam in our setup is 26.6 μm.Table 2 at FIG. 9 shows the relationship between the observed beam sizeand distance from focal point to the final plano-convex lens' holder.Beam size was characterized by measuring the incandescent surface areaof a 500 μm thick glass substrate. When focusing the laser onto theglass surface, the surface generated incandescence when a criticalenergy density was reached. During the laser beam size characterization,the circular coverslip attenuator was removed to allow higher energydensity. As the distance between glass surface and lens holderapproached 46 mm, power required to reach incandescence reduces until aminimal beam size of 28.93 μm was achieved. Overall, we have establisheda CO2 laser system that enables precise control of differentialevaporation onto liquid sample surface at a spatial resolution of 28.93μm.

Verification of key assumptions. The following conditions on dropletgeometry and evaporation rate may be satisfied:

$\begin{matrix}{{{{Droplet}\mspace{14mu} {height}\mspace{14mu} {profile}\text{:}\mspace{14mu} {h\left( {r,t} \right)}} = {{H(t)}\mspace{14mu}\left\lbrack {1 - \frac{r^{2}}{R^{2}(t)}} \right\rbrack}},} & {{Equation}\mspace{14mu} (14)} \\{{{Overall}\mspace{14mu} {evaporation}\mspace{14mu} {rate}\text{:}\mspace{14mu} \overset{.}{V}} = {{{\overset{.}{V}}_{o}\left( \frac{R(t)}{R_{o}} \right)}.}} & {{Equation}\mspace{14mu} (15)}\end{matrix}$

In Equation (14), the droplet height profile h(r,t) follows a paraboliccurve described by its center height H(t) and contact radius R(t) attime t. To verify Equation (14), droplet volume from both parabolicheight profile estimation, and its exact volume described by sphericalprofile were compared:

Estimated droplet volume from parabolic profile:

$\begin{matrix}{{{V(t)} = {\frac{\pi}{2}\mspace{14mu} {H(t)}{R^{2}(t)}}},} & {{Equation}\mspace{14mu} (16)}\end{matrix}$

Exact droplet volume from spherical profile:

$\begin{matrix}{{V(t)} = {\frac{\pi}{6}\mspace{14mu} {{{H(t)}\left\lbrack {{3 \star {R^{2}(t)}} + {H(t)}^{2}} \right\rbrack}.}}} & {{Equation}\mspace{14mu} (17)}\end{matrix}$

In FIG. 7, estimated (Equation 16) and exact droplet volumes (Equation17) of drying droplets (n=3) during laser-induced differentialevaporation were calculated and plotted. While the parabolic andspherical droplet volume differs initially, the two curves convergerapidly after 0.3 T/T₀. Convergence of the two curves indicatesagreement between actual droplet geometry and mathematicalapproximation. Therefore, Equation 14 applies to our empirical resultswhen T/T₀>0.3.

In Equation 15, the overall evaporation rate V is a linear function ofinitial evaporation rate {dot over (V)}_(o), R (t), and initial contactradius R_(o).2,3 Equation 15 is valid when the {dot over (V)} isproportional to contact radius R (t) and independent of contact angle θ,which can be describe by the following condition:

$\begin{matrix}{{\frac{\overset{.}{R}}{R} > \frac{\overset{.}{\theta}}{\theta}},} & {{Equation}\mspace{14mu} (18)}\end{matrix}$

where {dot over (R)} is the contact radius time derivative and {dot over(θ)} is contact angle time derivative. In FIG. 8, absolute values of

$\frac{\overset{.}{R}}{R}\mspace{14mu} {and}\mspace{14mu} \frac{\overset{.}{\theta}}{\theta}$

of drying droplets (n=3) during laser-induced differential evaporationwere calculated and plotted. At

T/T₀ = 0.3,

$\frac{\overset{.}{R}}{R}$

outpaces and continues to increase

$\frac{\overset{.}{\theta}}{\theta}$

while

$\frac{\overset{.}{\theta}}{\theta}$

continues to decrease. When the condition set forth in Equation 18 istrue, droplet evaporation can be contributed mainly to contact radiuschange and thus Equation 15 is also true. Therefore, Equation 15 appliesto our results when T/T_(o)>0.3.

Derivation of analytic solution for drying pattern of laser-induceddifferential evaporation. Given Eqs. 14-15 are valid approximation forlaser-induced differential evaporation process, {dot over (R)} can befound. When applied to Stokesian hydrodynamics, the principle isequivalent to minimization in energy dissipation, which is defined bythe Rayleighian

=Φ+{dot over (F)}, and the Onsager principle4 here is defined as:

$\begin{matrix}{{\frac{\partial\left( {\Phi + \overset{.}{F}} \right)}{\partial\overset{.}{R}} = 0},} & {{{Equation}\mspace{14mu} (19)}\mspace{14mu}}\end{matrix}$

where {dot over (F)} is the time derivative of the free energy of thesystem and Φ is the energy dissipation function. Define

$\theta_{e} = \sqrt{\frac{2\left( {\gamma_{LS} + \gamma_{LV} - \gamma_{sV}} \right)}{\gamma_{LV}}}$

as the equilibrium contact angle:

$\begin{matrix}{{\overset{.}{F} = {\gamma_{LV}\left\{ {{\left\lbrack {\frac{{- 16}V^{2}}{\pi \; R^{5}} + {\pi \; R\; \theta_{e}^{2}}} \right\rbrack \overset{.}{R}} + \frac{8V\overset{.}{V}}{\pi \; R^{4}}} \right\}}},} & {{Equation}\mspace{14mu} (20)} \\{{\Phi = {\left\lbrack {\frac{1}{2}{\int_{0}^{R}{{dr}\; 2\pi \; r\frac{3\eta}{h}v^{2}}}} \right\rbrack + {{\pi\zeta}_{cl}R{\overset{.}{R}}^{2}}}},} & {{Equation}\mspace{14mu} (21)}\end{matrix}$

where γ_(LV) is the liquid-vapor interfacial energy density, v is heightaveraged fluid velocity (v>0 for outward flow and v<0 for inward flow),and ζ_(cl) is inverse of mobility of the contact line (ζ_(cl)→∞ ifpinned contact line, ζ_(cl)→0 if free moving contact line). Defining thesolvent mass conservation equation (sign of the velocity: positive forleaving the center, negative towards the center):

$\begin{matrix}{{\frac{d}{dt}{\int_{0}^{r}{{dr}^{\prime}2\; \pi \; r^{\prime}{h\left( {r^{\prime},t} \right)}}}} = {{{- 2}\; \pi \; {{rv}\left( {r,t} \right)}{h\left( {r,t} \right)}} - {\int_{0}^{r}{{dr}^{\prime}2\; \pi \; r^{\prime}{J\left( {r^{\prime},t} \right)}}}}} & {{Equation}\mspace{14mu} (22)}\end{matrix}$

The evaporation flux J(r,t) [Vol/s−area] can be represented as the sumof two components:

J(r,t)=J _(i) +J _(d),  Equation (23)

where J_(i) is the evaporation rate due to isothermal evaporationbecause of laser heating, and J_(d) is the differential evaporation bythe focused laser beam. From the measured temperature profile of thedroplet under the focused CO2 laser beam and the fact that laser-inducedevaporation dries the droplet 10 times faster than the uniformly heateddroplet (to 50 C) by a hot plate, we can assume that J_(d)>>J_(i) andthe evaporation flux can be approximated as

$\begin{matrix}\begin{matrix}{{{J\left( {r,t} \right)} \sim J_{i}}\overset{\sim}{=}{{0\mspace{14mu} r} > a}} \\{{{J\left( {r,t} \right)} \sim J_{d}} = \frac{P}{\pi \; a^{2}}} \\{r < a}\end{matrix} & {{Equation}\mspace{14mu} (24)}\end{matrix}$

where α is the radius of the laser selected area. P is the laser-inducedevaporation flux and is proportional to the droplet surface temperaturein the laser irradiated area.

In our case, we only consider the case r>a to remove the coffee-ringeffect outside the laser spot. In a droplet with r>a, from Eqs. 22-10,

∫₀ ^(r) dr′2πr′J(r′,t)=P.  Equation (25)

From Eqs. 22 and 25,

$\begin{matrix}{{\frac{d}{dt}{\int_{0}^{r}{{dr}^{\prime}2\; \pi \; r^{\prime}{h\left( {r^{\prime},t} \right)}}}} = {{{- 2}\; \pi \; {{rv}\left( {r,t} \right)}{h\left( {r,t} \right)}} - {P.}}} & {{Equation}\mspace{14mu} (26)}\end{matrix}$

Each term in Equation 26 can be represented as below:

${{\frac{d}{dt}{\int_{0}^{r}{{dr}^{\prime}2\; \pi \; r^{\prime}{{H(t)}\left\lbrack {1 - \frac{r^{\prime 2}}{R^{2}(t)}} \right\rbrack}}}} = {{{\pi \; r^{2}\overset{.}{H}} - {\frac{\pi \; r^{4}}{2}\frac{d}{dt}\frac{H}{R^{2}}}} = {{\pi \; r^{2}\overset{.}{H}} - {\frac{\pi \; r^{4}}{2}\frac{\overset{.}{H}}{R^{2}}} + {\frac{2\; \pi \; r^{4}}{2}\frac{H}{R^{3}}\overset{.}{R}}}}},{{2\; \pi \; r\; {v\left( {r,t} \right)}{h\left( {r,t} \right)}} = {2\; \pi \; {{{rH}(t)}\left\lbrack {1 - \frac{r^{2}}{R^{2}(t)}} \right\rbrack}{v.}}}$

Substitute all the above into Equation 26 to derive Equation 27

$\begin{matrix}{{{{{\pi \; r^{2}\overset{.}{H}} - {\frac{\pi \; r^{4}}{2}\frac{\overset{.}{H}}{R^{2}}} + {\frac{2\; \pi \; r^{4}}{2}\frac{H}{R^{3}}\overset{.}{R}}} = {{{- 2}\; \pi \; {{{rH}(t)}\left\lbrack {1 - \frac{r^{2}}{R^{2}(t)}} \right\rbrack}v} - P}}{{2\; \pi \; {{{rH}(t)}\left\lbrack {1 - \frac{r^{2}}{R^{2}(t)}} \right\rbrack}v} + P} = {{{- \pi}\; r^{2}\overset{.}{H}} + {\frac{\pi \; r^{4}}{2}\frac{\overset{.}{H}}{R^{2}}} - {\frac{2\; \pi \; r^{4}}{2}\frac{H}{R^{3}}\overset{.}{R}}}}{{{Right}\text{-}{hand}\text{-}{side}} = {{- {\frac{\pi \; r^{2}\overset{.}{H}}{2}\left\lbrack {2 - \frac{r^{2}}{R^{2}}} \right\rbrack}} - {\pi \; r^{2}H\frac{\overset{.}{R}}{R}\left( \frac{r^{2}}{R^{2}} \right)}}}{{2\; \pi \; {{{rH}(t)}\left\lbrack {1 - \frac{r^{2}}{R^{2}(t)}} \right\rbrack}v} = {{- {\frac{\pi \; r^{2}\overset{.}{H}}{2}\left\lbrack {2 - \frac{r^{2}}{R^{2}}} \right\rbrack}} - {\pi \; r^{2}H\frac{\overset{.}{R}}{R}\left( \frac{r^{2}}{R^{2}} \right)} - P}}{{2\; \pi \; {{rH}(t)}v} = {{- {\frac{\pi \; r^{2}\overset{.}{H}}{2}\left\lbrack {1 + \frac{1}{\left( {1 - \frac{r^{2}}{R^{2}}} \right)}} \right\rbrack}} + {\pi \; r^{2}H{\frac{\overset{.}{R}}{R}\left\lbrack {1 - \frac{1}{\left( {1 - \frac{r^{2}}{R^{2}}} \right)}} \right\rbrack}} - \frac{P}{\left\lbrack {1 - \frac{r^{2}}{R^{2}}} \right\rbrack}}}} & {{Equation}\mspace{14mu} (27)}\end{matrix}$

Use the following relations:

$\begin{matrix}{\mspace{79mu} {{{V(t)} = {\frac{\pi}{2}{H(t)}{R^{2}(t)}}},\mspace{79mu} {\overset{.}{V} = {{\frac{\pi}{2}\overset{.}{H}R^{2}} + {\pi \; {RH}\overset{.}{R}}}},}} & {{Equation}\mspace{14mu} (28)} \\{{\frac{\overset{.}{V}}{V} = {\frac{{\frac{\pi}{2}\overset{.}{H}R^{2}} + {\pi \; {RH}\overset{.}{R}}}{\frac{\pi}{2}{HR}^{2}} = {\frac{\overset{.}{H}}{H} + \frac{2\overset{.}{R}}{R}}}};{\frac{\overset{.}{H}}{4H} = {\frac{\overset{.}{V}}{4V} - {\frac{\overset{.}{R}}{2R}.}}}} & {{Equation}\mspace{14mu} (29)}\end{matrix}$

We can represent v:

$\begin{matrix}{{v = {{- {\frac{r\overset{.}{H}}{4H}\left\lbrack {1 + \frac{1}{\left( {1 - \frac{r^{2}}{R^{2}}} \right)}} \right\rbrack}} + {\frac{r}{2}{\frac{\overset{.}{R}}{R}\left\lbrack {1 - \frac{1}{\left( {1 - \frac{r^{2}}{R^{2}}} \right)}} \right\rbrack}} - \frac{P}{2\; \pi \; {{rH}\left\lbrack {1 - \frac{r^{2}}{R^{2}}} \right\rbrack}}}},} & {{Equation}\mspace{14mu} (30)}\end{matrix}$

By reorganizing the terms, we obtain

$\begin{matrix}{{v = {{r\left\lbrack {\frac{\overset{.}{R}}{R} - \frac{\overset{.}{V}}{4V}} \right\rbrack} - {\frac{r}{\left( {1 - \frac{r^{2}}{R^{2}}} \right)}\frac{\overset{.}{V}}{4V}} - \frac{P}{2\; \pi \; {r\left\lbrack {1 - \frac{r^{2}}{R^{2}}} \right\rbrack}H}}},} & {{Equation}\mspace{14mu} (31)}\end{matrix}$

Substitute Equation 31 into Equation 21,

$\begin{matrix}{\Phi = {\left\lbrack {\frac{1}{2}{\int_{0}^{R}{{dr}\; 2\; \pi \; r\frac{3\eta}{h}\left\{ {{r\left\lbrack {\frac{\overset{.}{R}}{R} - \frac{\overset{.}{V}}{4V}} \right\rbrack} - {\frac{r}{\left( {1 - \frac{r^{2}}{R^{2}}} \right)}\frac{\overset{.}{V}}{4V}} - \frac{P}{2\; \pi \; {r\left\lbrack {1 - \frac{r^{2}}{R^{2}}} \right\rbrack}H}} \right\}^{2}}}} \right\rbrack {\quad{\quad{\quad{\quad{+ {\quad{\quad{{{\quad\quad}\pi \; \zeta_{cl}R{{\overset{.}{R}}^{2}.\Phi}} = {{\frac{1}{2}{\int_{0}^{R}{{dr}\; 2\; \pi \; r^{3}\frac{3\eta}{h}\left\{ {\left\lbrack {\frac{\overset{.}{R}}{R} - \frac{\overset{.}{V}}{4V}} \right\rbrack^{2} - {\frac{1}{\left( {1 - \frac{r^{2}}{R^{2}}} \right)}{\frac{\overset{.}{V}}{2V}\left\lbrack {\frac{\overset{.}{R}}{R} - \frac{\overset{.}{V}}{4V}} \right\rbrack}} + {\frac{1}{\left( {1 - \frac{r^{2}}{R^{2}}} \right)^{2}}\left( \frac{\overset{.}{V}}{4V} \right)^{2}}} \right\}}}} - {\quad{\left\lbrack {\int_{0}^{R}{{dr}\frac{3\eta}{h}\left\{ {{r\left\lbrack {\frac{\overset{.}{R}}{R} - \frac{\overset{.}{V}}{4V}} \right\rbrack} - {\frac{r}{\left( {1 - \frac{r^{2}}{R^{2}}} \right)}\frac{\overset{.}{V}}{4V}}} \right\} \frac{P}{\left\lbrack {1 - \frac{r^{2}}{R^{2}}} \right\rbrack H}}} \right\rbrack + {\quad{\left\lbrack {\frac{1}{2}{\int_{0}^{R}{{dr}\frac{1}{2\; \pi \; r}\frac{3\; \eta \; P^{2}}{\left\lbrack {1 - \frac{r^{2}}{R^{2}(t)}} \right\rbrack^{3}}}}} \right\rbrack + {\pi \; \zeta_{cl}R{\overset{.}{R}}^{2}}}}}}}}}}}}}}}}} & {{Equation}\mspace{14mu} (32)}\end{matrix}$

To avoid singularity in energy dissipation at the contact line in 32, weuse a molecular cutoff length to be in the order of 10⁻⁶R and define

$\epsilon = {{\lim\limits_{r\rightarrow{{({1 - 10^{- 6}})}R}}\left( {1 - \frac{r^{2}}{R^{2}}} \right)} = {2 \times {10^{- 6}.}}}$

The first term in Equation 32 is

${\frac{1}{2}{\int_{0}^{R}{{dr}\; 2\; \pi \; r^{3}\frac{3\eta}{h}\left\{ {\left\lbrack {\frac{\overset{.}{R}}{R} - \frac{\overset{.}{V}}{4V}} \right\rbrack^{2} - {\frac{1}{\left( {1 - \frac{r^{2}}{R^{2}}} \right)}{\frac{\overset{.}{V}}{2V}\left\lbrack {\frac{\overset{.}{R}}{R} - \frac{\overset{.}{V}}{4V}} \right\rbrack}} + {\frac{1}{\left( {1 - \frac{r^{2}}{R^{2}}} \right)^{2}}\left( \frac{\overset{.}{V}}{4V} \right)^{2}}} \right\}}}} = {{{\frac{3\; \pi^{2}\eta \; R^{4}}{4V}\left\lbrack {{- {\ln (\epsilon)}} - 1} \right\rbrack}\left( {\overset{.}{R} - \frac{R\overset{.}{V}}{4V}} \right)^{2}} - {\frac{3\; \pi^{2}\eta \; R^{4}}{4V}\left( \frac{R\overset{.}{V}}{4V} \right){\left( {\overset{.}{R} - \frac{R\overset{.}{V}}{4V}} \right)\left\lbrack {\frac{1}{\epsilon} + {\ln \; \epsilon} - 1} \right\rbrack}} + {\frac{3\; \pi^{2}\eta \; R^{4}}{4V}{\left( \frac{R\overset{.}{V}}{4V} \right)^{2}\left\lbrack {\frac{1}{2\; \epsilon^{2}} - \frac{1}{\epsilon} + \frac{1}{2}} \right\rbrack}}}$

Also, the second term in Equation 32 is

$\left\lbrack {\int_{0}^{R}{{dr}\frac{3\; \eta}{h}\left\{ {{r\left\lbrack {\frac{\overset{.}{R}}{R} - \frac{\overset{.}{V}}{4V}} \right\rbrack} - {\frac{r}{\left( {1 - \frac{r^{2}}{R^{2}}} \right)}\frac{\overset{.}{V}}{4V}}} \right\} \frac{P}{\left\lbrack {1 - \frac{r^{2}}{R^{2}}} \right\rbrack H}}} \right\rbrack = {{\frac{3\; \pi^{2}\eta \; R^{4}}{4V}\frac{{PR}^{2}}{2V}{\left( {\frac{\overset{.}{R}}{R} - \frac{\overset{.}{V}}{4V}} \right)\left\lbrack {\frac{1}{\epsilon} - 1} \right\rbrack}} - {\frac{3\pi^{2}\eta \; R^{4}}{4V}\frac{{PR}^{2}}{4V}{\left( \frac{\overset{.}{V}}{4V} \right)\left\lbrack {\frac{1}{\epsilon^{2}} - 1} \right\rbrack}}}$

The third term in Equation 32 is not expanded here since it does nothave a {dot over (R)} component and thus will not be contributing to ourderivation of Equation 33.

Using Eqs. 19, 20, and 32, then

$\begin{matrix}{{{{\gamma_{LV}\left\lbrack {\frac{{- 16}V^{2}}{\pi \; R^{5}} + {\pi \; R\; \theta_{e}^{2}}} \right\rbrack} + {\frac{3\; \pi^{2}\eta \; R^{4}}{2V}{\left( {\overset{.}{R} - \frac{R\overset{.}{V}}{4V}} \right)\left\lbrack {{- {\ln (\epsilon)}} - 1} \right\rbrack}} - {\frac{3\; \pi^{2}\eta \; R^{4}}{4V}{\left( \frac{R\overset{.}{V}}{2V} \right)\left\lbrack {\frac{1}{\epsilon} + {\ln \; \epsilon} - 1} \right\rbrack}} - {\frac{3\pi^{2}\eta \; R^{4}}{4V}{\frac{PR}{2V}\left\lbrack {\frac{1}{\epsilon} - 1} \right\rbrack}} + {2\; \pi \; \zeta_{cl}R\overset{.}{R}}} = 0}\;} & {{Equation}\mspace{14mu} (33)}\end{matrix}$

Using the approximation: {dot over (V)}˜−P, Equation 33 can be writtenas

${{\gamma_{LV}\left\lbrack {\frac{{- 16}V^{2}}{\pi \; R^{5}} + {\pi \; R\; \theta_{e}^{2}}} \right\rbrack} + {\frac{3\; \pi^{2}\eta \; R^{4}}{2V}{\left( {\overset{.}{R} - \frac{R\overset{.}{V}}{4V}} \right)\left\lbrack {{- {\ln (\epsilon)}} - 1} \right\rbrack}} + {\frac{3\; \pi^{2}\eta \; R^{4}}{4V}{\left( \frac{RP}{2V} \right)\left\lbrack {\ln \; \epsilon} \right\rbrack}} + {2\; {\pi\zeta}_{cl}R\overset{.}{R}}} = 0$

Define

$\begin{matrix}{{{{\left( {1 + \frac{4\; {\pi\zeta}_{cl}{RV}}{3\; \pi^{2}\eta \; R^{4}C}} \right)\overset{.}{R}} - {\left( \frac{RP}{4V} \right)\left( {1 + \frac{1}{C}} \right)}} = {\frac{R\overset{.}{V}}{4V} + {\frac{2V}{3\; \pi^{2}\eta \; {CR}^{4}}{\gamma_{LV}\left\lbrack {\frac{16V^{2}}{{\pi \; R^{5}}\;} = {\pi \; R\; \theta_{e}^{2}}} \right\rbrack}}}},} & {{Equation}\mspace{14mu} (34)}\end{matrix}$

Use the relations:

${\theta (t)} = \frac{2{H(t)}}{R(t)}$

and Equation 16,

${\frac{{16V^{2}}\;}{\pi \; R^{5}} = \frac{4\; \pi \; H^{2}}{R}},{\left\lbrack {\frac{16V^{2}}{\pi \; R^{5}} - {\pi \; R\; \theta_{e}^{2}}} \right\rbrack = {{\frac{4\; \pi \; H^{2}}{R} - {\pi \; R\; \theta_{e}^{2}}} = {{\pi \; {R\left( {\theta^{2} - \theta_{e}^{2}} \right)}\frac{2V}{3\; \pi^{2}\eta \; {CR}^{4}}} = {\frac{\pi \; {HR}^{2}}{3\; \pi^{2}\eta \; {CR}^{4}} = {\frac{\pi \; H}{3\; \pi^{2}\eta \; {CR}^{2}} = \frac{\theta}{6\; \pi \; \eta \; {CR}}}}}}},$

thus the 2nd term in the RHS of Equation 34 can be written as

${\frac{2V}{3\; \pi^{2}\eta \; {CR}^{4}}{\gamma_{LV}\left\lbrack {\frac{16\; V^{2}}{\pi \; R^{5}} = {\pi \; R\; \theta_{e}^{2}}} \right\rbrack}} = {\frac{{\theta\gamma}_{LV}}{6\; \eta \; C}\left( {\theta^{2} - \theta_{e}^{2}} \right)}$

Also the term in the LHS of Equation 34:

$\frac{4\; \pi \; \zeta_{cl}{RV}}{3\; \pi^{2}\eta \; R^{4}C}$

can be simplified as

$\begin{matrix}{\frac{4\; \zeta_{cl}{RV}}{3\; \pi \; \eta \; R^{4}C} = {\frac{\zeta_{cl}\theta}{3\; \eta \; C} \equiv k_{cl}}} & {{Equation}\mspace{14mu} (35)}\end{matrix}$

As disclosed above, k_(cl) denotes the ratio of contact line friction tohydrodynamic friction. Here, k_(cl) can be assumed to be atime-independent material parameter that is determined by the dropletand the substrate.

We can simplify Equation 34 to

$\begin{matrix}{{{\left( {1 + k_{cl}} \right)\overset{.}{R}} = {\frac{R\overset{.}{V}}{4V} + {\frac{\gamma_{LV}}{6\; \eta \; C}{\theta \left( {\theta^{2} - \theta_{e}^{2}} \right)}} + {\left( \frac{RP}{4V} \right)\left( {1 + \frac{1}{C}} \right)}}},} & {{Equation}\mspace{14mu} (36)}\end{matrix}$

Consider solute at r_(o) at t=0. As the solvent evaporates, such asolute is convected by the fluid. {tilde over (r)}(r_(o),t) is theheight-averaged position of such a solute at t. Ignoring diffusion,solute moves at the same speed as the fluid as long as it is in thedroplet (i.e. {tilde over (r)}(r_(o),t)<R(t)). Hence, from Equation 31,

${\left( \overset{.}{\overset{\sim}{r}} \right)\left( {r_{o},t} \right)} = {{v\left\lbrack {{\overset{\sim}{r}\left( {r_{o},t} \right)},t} \right\rbrack} = {{\overset{\sim}{r}\left\lbrack {\frac{\overset{.}{R}}{R} - \frac{\overset{.}{V}}{4V}} \right\rbrack} - {\frac{\overset{\sim}{r}}{\left( {1 - \frac{{\overset{\sim}{r}}^{2}}{R^{2}}} \right)}\frac{\overset{.}{V}}{4V}} - {\frac{P}{2\; \pi \; {\overset{\sim}{r}\left\lbrack {1 - \frac{{\overset{\sim}{r}}^{2}}{R^{2}}} \right\rbrack}H}.}}}$

Using the approximate relation: P={dot over (V)}, the above equation canbe written as

$\begin{matrix}{{\left( \overset{.}{\overset{\sim}{r}} \right)\left( {r_{o},t} \right)} = {{v\left\lbrack {{\overset{\sim}{r}\left( {r_{o},t} \right)},t} \right\rbrack} = {{\overset{\sim}{r}\left\lbrack {\frac{\overset{.}{R}}{R} - \frac{\overset{.}{V}}{4V}} \right\rbrack} - \frac{P}{2\; \pi \; \overset{\sim}{r}H}}}} & {{Equation}\mspace{14mu} (37)}\end{matrix}$

To help simplifying our final expression, the following time scales andK_(ev) may be used:

$\begin{matrix}{{\tau_{ev} = {\frac{V_{o}}{{\overset{.}{V}}_{o}} = {- \frac{V_{o}}{{\overset{.}{V}}_{o}}}}},{\tau_{re} = \frac{\eta \; V_{0}^{\frac{1}{3}}}{\gamma_{LV}\theta_{3}^{3}}},{K_{ev} = {\frac{\tau_{re}}{\tau_{ev}}.}}} & {{Equation}\mspace{14mu} (38)}\end{matrix}$

where τ_(ev) represents the characteristic time for the droplet to drycompletely and τ_(re) represents the characteristic time to relax to itsequilibrium contact angle. Assume K_(ev)>>1 for concentrated polymer(i.e. high viscosity due to enrichment) or strong laser enhancedevaporation effect (i.e. K_(ev)>>1). Equation 36 can be approximated as

$\begin{matrix}{{\overset{.}{R} \sim {\frac{R\overset{.}{V}}{4\left( {1 + k_{cl}} \right)V} + {\frac{RP}{4{V\left( {1 + k_{cl}} \right)}}\left( {1 + \frac{1}{c}} \right)}}},} & {{Equation}\mspace{14mu} (39)} \\{\frac{\overset{.}{V}}{4V} = {{\left( {1 + k_{cl}} \right)\frac{\overset{.}{R}}{R}} - {\frac{P}{4V}\left( {1 + \frac{1}{c}} \right)}}} & {{Equation}\mspace{14mu} (40)}\end{matrix}$

From Eqs. 38 and 40, for {tilde over (r)}>a, we can write Equation 37as:

$\begin{matrix}{{{\left( \overset{.}{\overset{\sim}{r}} \right)\left( {r_{0},t} \right)} = {{v\left\lbrack {{\overset{\sim}{r}\left( {r_{0}t} \right)},t} \right\rbrack} = {{{- k_{cl}}{\overset{\sim}{r}\left( {r_{0},t} \right)}\frac{\overset{.}{R}}{R}} + {\frac{\overset{\sim}{r}P}{4V}\left( {1 + \frac{1}{c}} \right)} - \frac{P}{2\pi \overset{\sim}{r}H}}}}\mspace{20mu} {{{{for}\mspace{14mu} \overset{\sim}{r}} > a},\mspace{20mu} {{\frac{\left( \overset{.}{\overset{\sim}{r}} \right)}{\overset{\sim}{r}}\left( {r_{0},t} \right)} = {{{- k_{cl}}\frac{\overset{.}{R}}{R}} + {\frac{P}{4V}\left( {1 + \frac{1}{c}} \right)} - \frac{P}{2\pi {\overset{\sim}{r}}^{2}H}}},}} & {{Equation}\mspace{14mu} (41)} \\{\mspace{79mu} {{{\frac{\left( \overset{.}{\overset{\sim}{r}} \right)}{\overset{\sim}{r}}\left( {r_{0},t} \right)} = {{{- k_{cl}}\frac{\overset{.}{R}}{R}} - {\frac{P}{2\pi \; H}\left\lbrack {\frac{1}{{\overset{\sim}{r}}^{2}} - {\frac{1}{R^{2}}\left( {1 + \frac{1}{c}} \right)}} \right\rbrack}}},}} & {{Equation}\mspace{14mu} (42)}\end{matrix}$

Note that because of the negative sign in front of the

$\frac{\overset{.}{R}}{R}$

term, the solute moves in the opposite rate as the droplet when thelaser induced evaporation rate is turned off, revealing the coffee-ringeffect. The greater is the coefficient k_(cl) that is related to liquidviscosity and contact line friction, the more serious the coffee-ringeffect becomes. However, when the laser-induced evaporation is set at anappropriate level (to be determined next), and provided the solute atposition r_(o) initially (t=0) precipitates at t=t_(d) at droplet edge({tilde over (r)}(r_(o),t_(d))=R), the velocity of solute ({tilde over({dot over (r)})}(r_(o),t_(d)) can be in the same direction as {dot over(R)} but at a higher magnitude, yielding a condition that contradictsthe presumption that the solute precipitates at the edge of the droplet.Therefore, when the laser induced differential evaporation rate reachesan appropriate level, solute with its initial position r_(o) will notprecipitate at the edge of the droplet, thus removing the coffee-ringeffect.

Since Equation 42 is a function of the laser-induced evaporation rateP(t), we can design P(t) to approximately yield the following solutionthat will produce the deposition pattern to be discussed next:

$\begin{matrix}{{{\overset{\sim}{r}\left( {r_{0},t} \right)} = {r_{0}\left( \frac{R}{R_{0}} \right)}^{- {\lbrack{k_{cl} - G}\rbrack}}},} & {{Equation}\mspace{14mu} (43)}\end{matrix}$

with r_(o) being the initial solute position (at t=0) within the dropletand {tilde over (r)} being the solute position at time “t”. InsertingEquation 43 into Equation 42, we can obtain

$\begin{matrix}{{{{{GR}\overset{.}{R}} = {- {\frac{P}{2\pi \; H}\left\lbrack {\frac{R^{2}}{{\overset{\sim}{r}}^{2}} - \left( {1 + \frac{1}{c}} \right)} \right\rbrack}}},{{{since}\mspace{14mu} \frac{R^{2}}{{\overset{\sim}{r}}^{2}}} = {\frac{R^{2}}{r_{0}^{2}}\frac{R_{0}^{2{\lbrack{G - k_{cl}}\rbrack}}}{R^{2{\lbrack{G - k_{cl}}\rbrack}}}}}}{{{GR}\overset{.}{R}} = {- {{\frac{P}{2\pi \; H}\left\lbrack {{\frac{R^{2}}{r_{0}^{2}}\frac{R_{0}^{2{\lbrack{G - k_{cl}}\rbrack}}}{R^{2{\lbrack{G - k_{cl}}\rbrack}}}} - \left( {1 + \frac{1}{c}} \right)} \right\rbrack}.}}}} & {{Equation}\mspace{14mu} (44)}\end{matrix}$

By choosing G>>k_(cl)+1 to remove coffee-ring and keeping in mind thatGR{dot over (R)}<0 since {dot over (R)}<0, then

$\begin{matrix}{{{GR}\overset{.}{R}} \sim {- {\frac{P}{2\pi \; H}\left\lbrack {{\frac{R^{2}}{r_{0}^{2}}\frac{R_{0}^{2{\lbrack{G - k_{cl}}\rbrack}}}{R^{2{\lbrack{G - k_{cl}}\rbrack}}}} - \left( {1 + \frac{1}{c}} \right)} \right\rbrack}} \sim {- {{\frac{P}{2\pi \; H}\left\lbrack {\frac{R^{2}}{r_{0}^{2}}\frac{R_{0}^{2{\lbrack{G - k_{cl}}\rbrack}}}{R^{2{\lbrack{G - k_{cl}}\rbrack}}}} \right\rbrack}.}}} & {{Equation}\mspace{14mu} (45)}\end{matrix}$

To assure that

${{\frac{R^{2}}{r_{0}^{2}}\frac{R_{0}^{2{\lbrack{G - k_{cl}}\rbrack}}}{R^{2{\lbrack{G - k_{cl}}\rbrack}}}}\left( {1 + \frac{1}{c}} \right)},$

we must have

$\left( \frac{R_{0}}{R} \right)^{2{({G - k_{cl} - 1})}}\left( {1 + \frac{1}{c}} \right)$

and thus we can set a condition for G in Equation 43 as

$\begin{matrix}{G > {k_{cl} + 1 + {\frac{\ln \mspace{14mu} 1.08}{2\mspace{14mu} {\ln \left( \frac{R_{0}}{R} \right)}}.}}} & {{Equation}\mspace{14mu} (46)}\end{matrix}$

While R→R₀ leads to G→∞, please note that our parabolic volume profileis only valid for timeframe T/T₀>0.3. From FIG. 3, In

${{\frac{R_{0}}{R} > {0.15\mspace{14mu} {at}\mspace{14mu} {T/T_{0}}}} = 0.4},$

and thus G>k_(cl)+1.27. To prevent the solute that is initially locatedat ro from precipitation, we set the laser-induced evaporation rate tosatisfy the following condition:

${P(t)} > {{- {GR}}{\overset{.}{R}\left( {2\pi \; H} \right)}\mspace{11mu} \left( \frac{R}{R_{0}} \right)^{2{({G - k_{cl} - 1})}}\left( \frac{r_{0}^{2}}{R_{0}^{2}} \right)}$

Since r_(o)≤R_(o) and R≤R_(o), we can safely set the laser power to bethe following to assure that the real solution of Equation 45 is boundedby the expression of Equation 43.

$\begin{matrix}{{{P(t)} > {{- {GR}}{\overset{.}{R}\left( {2\pi \; H} \right)}}} = {{GH}\frac{d}{dt}\left( {\pi \; R^{2}} \right)}} & {{Equation}\mspace{14mu} (47)}\end{matrix}$

Assume the solute eventually precipitates at position {tilde over(r)}(r_(o),t_(d))=R(t_(d)), from Equation 43,

$\begin{matrix}{{\overset{\sim}{r}\left( {r_{0},t_{d}} \right)} = {{R\left( t_{d} \right)} = {\left( r_{0} \right)^{\frac{1}{1 + k_{cl} - G}}{\left( R_{0} \right)^{\frac{k_{cl} - G}{1 + k_{cl} - G}}.}}}} & {{Equation}\mspace{14mu} (48)}\end{matrix}$

Inserting Equation 48 into the expression of drying pattern depositdensity,

$\begin{matrix}{{{u\left( {\overset{\sim}{r}\left( {r_{0},t} \right)} \right)} = {{h\left( {r_{0},0} \right)}Ø_{0}\frac{r_{0}}{\overset{\sim}{r}}\left( \frac{d\overset{\sim}{r}}{{dr}_{0}} \right)^{- 1}}},} & {{Equation}\mspace{14mu} (49)}\end{matrix}$

where Ø_(o) is initial solute concentration (at t=0, R=R_(o)) and H_(o)is the initial droplet height at t=0, and representing r_(o) in terms of{tilde over (r)}, Equation 49 becomes

$\begin{matrix}{{{u\left( \overset{\sim}{r} \right)} = {Ø_{0}{H_{0}\left( {1 + k_{cl} - T} \right)}{\left( \frac{\overset{\sim}{r}}{R_{0}} \right)^{2{\lbrack{k_{cl} - G}\rbrack}}\left\lbrack {1 - \left( \frac{\overset{\sim}{r}}{R_{0}} \right)^{2{\lbrack{1 + k_{cl} - G}\rbrack}}} \right\rbrack}}},} & {{Equation}\mspace{14mu} (50)}\end{matrix}$

which is our final solution for drying pattern of laser-induceddifferential evaporation.

FIG. 7 depicts an example of estimated droplet volume from parabolicprofile vs. exact droplet volume from spherical profile of dryingdroplets (n=3) during laser-induced differential evaporation.

FIG. 8 depicts {dot over (R)}/R and {dot over (θ)}/θ for of dryingdroplets (n=3) during laser-induced differential evaporation.

Table 2 in FIG. 9 depicts an example of observed beam size vs. distanceto lens.

Based on the description herein, an apparatus that is suitable forcausing microdroplet evaporation may be implemented to include a lightsource, a substrate to hold a microdroplet at a location, and a focusingmodule to focus the light source at an apex of the microdroplet, whereina beam waist of the focused light source has a diameter less than adiameter of the microdroplet. For example, microdroplets may have adiameter of less than 100 micrometers. For example, beam waist mayrepresent the narrowest beam width during the operation. The focusingapparatus may be designed and controlled to achieve this beam waist forthe microdroplet evaporation apparatus.

For example, the focusing module may be implemented using an attenuatedlaser and as described with reference to FIGS. 1A-1C. For example, theplano-convex lens 130 or another suitable light focusing mechanism,e.g., an alignment microscope as depicted in FIG. 1A, may be used forimplementing the focusing module. In some implementation, the focusingmodule causes illumination by the light source of a plurality of wellsincluding the well at the same time or nearly the same time.

In some embodiments, e.g., described in FIGS. 1A-1C, the apparatusfurther includes an x-y mechanical translation apparatus configured toselectively translate the substrate to one of a plurality of wells withcorresponding microdroplets including the well. For example, themicroarray depicted in FIG. 1C may be used for providing the mechanicaltranslation. For example, the alignment microscope may be used tocorrectly align with the microdroplet. As further described in thepresent document, the apparatus causes a drying of the microdropletleaving a deposition of solids that were previously dissolved in themicrodroplet before the microdroplet was dried. The deposition of solidsis uniform or nearly-uniform (e.g., within 10% tolerance) across thedried microdroplet.

As disclosed in the present document, the apparatus may further includethe following features:

In some embodiments, the microdroplet evaporation apparatus causes adrying of the microdroplet leaving a deposition of solids that werepreviously dissolved in the microdroplet before the microdroplet wasdried.

In some embodiments, the deposition of solids is approximately the samesize as the microdroplet.

In some embodiments, the deposition of solids includes deposition ofwater-soluble molecules including one or more of nucleic acids,proteins, inks, and other small molecules.

In some embodiments, the light source is a laser.

In some embodiments, the laser is a CO2 laser.

In some embodiments, the beam waist of the CO2 laser is approximately25-35 microns.

In some embodiments, a beam of the CO2 laser is focused on the apex ofthe microdroplet to generate a differential evaporation.

In some embodiments, a differential evaporative flux profile has amaximum at the apex of the microdroplet.

In some embodiments, the beam focused at the apex of the microdropletcauses a uniform or nearly uniform deposition of solids when themicrodroplet is dried. 14. The microdroplet evaporation apparatus ofclaim 1, wherein the microdroplet is an aqueous solution droplet with adroplet diameter of between 100 μm and 1.5 mm.

In some embodiments, the well is approximately 100 μm in diameter.

In some embodiments, the microdroplet has a volume between 1-10microliters.

Based on the description herein, a method of evaporating a microdropletmay be performed to include: holding, by a substrate, a microdroplet ata location, illuminating, by a light source, the microdroplet, andfocusing the light source at an apex of the microdroplet to cause adrying of the microdroplet, wherein a beam waist of the focused lightsource has a diameter less than a diameter of the microdroplet.

In some embodiments, the drying of the microdroplet leaves a depositionof solids that were previously dissolved in the microdroplet before themicrodroplet was dried. In some embodiments, the deposition of solids isuniform or nearly uniform across the dried microdroplet.

With respect to the above described apparatus and method, in someembodiments, the deposition of solids includes deposition ofwater-soluble molecules including one or more of nucleic acids,proteins, inks, and other small molecules.

Although a few variations have been described in detail above, othermodifications or additions are possible. In particular, further featuresand/or variations may be provided in addition to those set forth herein.Moreover, the example embodiments described above may be directed tovarious combinations and subcombinations of the disclosed featuresand/or combinations and subcombinations of several further featuresdisclosed above. In addition, the logic flow depicted in theaccompanying figures and/or described herein does not require theparticular order shown, or sequential order, to achieve desirableresults. Other embodiments may be within the scope of the followingclaims.

Similarly, while operations are depicted in the drawings in a particularorder, this should not be understood as requiring that such operationsbe performed in the particular order shown or in sequential order, orthat all illustrated operations be performed, to achieve desirableresults. Moreover, the separation of various system components in theembodiments described in this patent document should not be understoodas requiring such separation in all embodiments.

Only a few implementations and examples are described and otherimplementations, enhancements and variations can be made based on whatis described and illustrated in this patent document.

What is claimed is:
 1. A microdroplet evaporation apparatus, comprising:a light source; a substrate to hold a microdroplet at a location; and afocusing module configured to focus the light source at an apex of themicrodroplet, such that a beam waist of the focused light source has adiameter less than a diameter of the microdroplet.
 2. The microdropletevaporation apparatus of claim 1, wherein the focusing module causesillumination by the light source of a plurality of wells including thewell at the same time or nearly the same time.
 3. The microdropletevaporation apparatus of claim 1, further comprising: an x-y mechanicaltranslation apparatus configured to selectively translate the substrateto one of a plurality of wells with corresponding microdropletsincluding the well.
 4. The microdroplet evaporation apparatus of claim1, wherein the microdroplet evaporation apparatus causes a drying of themicrodroplet leaving a deposition of solids that were previouslydissolved in the microdroplet before the microdroplet was dried.
 5. Themicrodroplet evaporation apparatus of claim 4, wherein the deposition ofsolids is uniform or nearly uniform across the dried microdroplet. 6.The microdroplet evaporation apparatus of claim 4, wherein thedeposition of solids is approximately the same size as the microdroplet.7. The microdroplet evaporation apparatus of claim 4, wherein thedeposition of solids includes deposition of water-soluble moleculesincluding one or more of nucleic acids, proteins, inks, and other smallmolecules.
 8. The microdroplet evaporation apparatus of claim 1, whereinthe light source is a laser.
 9. The microdroplet evaporation apparatusof claim 1, wherein the laser is a CO₂ laser.
 10. The microdropletevaporation apparatus of claim 9, wherein the beam waist of the CO₂laser is approximately 25-35 microns.
 11. The microdroplet evaporationapparatus of claim 9, wherein a beam of the CO₂ laser is focused on theapex of the microdroplet to generate a differential evaporation.
 12. Themicrodroplet evaporation apparatus of claim 11, wherein a differentialevaporative flux profile has a maximum at the apex of the microdroplet.13. The microdroplet evaporation apparatus of claim 12, wherein the beamfocused at the apex of the microdroplet causes a uniform or nearlyuniform deposition of solids when the microdroplet is dried.
 14. Themicrodroplet evaporation apparatus of claim 1, wherein the microdropletis an aqueous solution droplet with a droplet diameter of between 100 μmand 1.5 mm.
 15. The microdroplet evaporation apparatus of claim 1,wherein the well is approximately 100 μm in diameter.
 16. Themicrodroplet evaporation apparatus of claim 1, wherein the microdroplethas a volume between 1-10 microliters.
 17. A method of evaporating amicrodroplet, comprising: holding, by a substrate, a microdroplet at alocation; illuminating, by a light source, the microdroplet; andfocusing the light source at an apex of the microdroplet to cause adrying of the microdroplet, wherein a beam waist of the focused lightsource has a diameter less than a diameter of the microdroplet.
 18. Themethod of evaporating a microdroplet of claim 17, wherein the drying ofthe microdroplet leaves a deposition of solids that were previouslydissolved in the microdroplet before the microdroplet was dried.
 19. Themethod of evaporating a microdroplet of claim 18, wherein the depositionof solids is uniform or nearly uniform across the dried microdroplet.20. The method of evaporating a microdroplet of claim 18, wherein thedeposition of solids includes deposition of water-soluble moleculesincluding one or more of nucleic acids, proteins, inks, and other smallmolecules.